Theorem sucexeloni 7749

Description: If the successor of an ordinal number exists, it is an ordinal number. This variation of onsuc 7751 does not require ax-un 7675. (Contributed by BTernaryTau, 30-Nov-2024.) (Proof shortened by BJ, 11-Jan-2025.)

Assertion

Ref	Expression
sucexeloni	⊢ ((𝐴 ∈ On ∧ suc 𝐴 ∈ 𝑉) → suc 𝐴 ∈ On)

Proof of Theorem sucexeloni

Step	Hyp	Ref	Expression
1		eloni 6321	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordsuci 7748	. . 3 ⊢ (Ord 𝐴 → Ord suc 𝐴)
3	1, 2	syl 17	. 2 ⊢ (𝐴 ∈ On → Ord suc 𝐴)
4		elex 3459	. 2 ⊢ (suc 𝐴 ∈ 𝑉 → suc 𝐴 ∈ V)
5		elong 6319	. . 3 ⊢ (suc 𝐴 ∈ V → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
6	5	biimparc 479	. 2 ⊢ ((Ord suc 𝐴 ∧ suc 𝐴 ∈ V) → suc 𝐴 ∈ On)
7	3, 4, 6	syl2an 596	1 ⊢ ((𝐴 ∈ On ∧ suc 𝐴 ∈ 𝑉) → suc 𝐴 ∈ On)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  Vcvv 3438  Ord word 6310  Oncon0 6311  suc csuc 6313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-tr 5203  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-ord 6314  df-on 6315  df-suc 6317

This theorem is referenced by:  onsuc  7751  1on  8407  2on  8408